Suppose that the metric space $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'_i)$ for $i=1,2, \cdots , n$. Show that the product metric spaces $X = \prod_{i=1}^nX_i$ and $Y= \prod_{i=1}^nY_i$ are topologically equivalent.
I know that since $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'_i)$ for $i=1, \cdots , n$, then there exists a homeomorphism $f: (X_i,d_i) \to (Y_i,d'_i)$ from $(X_i, d_i)$ onto $(Y_i,d'_i)$ for which $f$ and its inverse $f^{-1}$ are both continuous. This is the definition of topological equivalence given in my book.
However, I am confused as to which homeomorphism to choose. I am given that it already exists, so do I just pick one, verify it is onto and one-to-one, and check for continuity of itself and its inverse? If this is the case, then how do I define this function in the first place?
Thank you very much for your time. Any suggestions and hints are most certainly appreciated. By the way, the textbook being used here is Principles of Topology by Fred H. Croom.